Wash your veggies when you buy them, dry them thoroughly, and store them in a sanitary environment; they'll last much longer this way.
I regularly keep my organic veggies for weeks at a time before cooking them with no molds or rotting.
It also helps to buy from a growers market where things haven't been in cold storage for months before getting to the store.
I wash with plain water and a little bit of citric acid or vinegar.
Keep (unused) silica gel kitty litter in a mesh bag in your fridge to lower the relative humidity, I lived in a swamp for a while too :/
Edit: dry your silica gel in the oven every couple weeks, once it looks like it's losing its transparency.